The Balance

Why balance?

In a duplicate bridge match you usually don't meet all other pairs. With 12 pairs for example, you need 11 rounds to meet all opponents. A club session often only has 6 rounds. After two sessions you then have possibly met all other pairs, but at least one pair you met twice. In short, a balanced system of encounters is often impossible.

Further, which opponents you play against is not the only factor. When you play a certain board as NS, all NS players on this board are indirectly your opponents. If they have a better result than you have, your score goes down and vice versa. On the other hand all EW pairs are your allies. It is good for your score if they do well.

Now the aim of a good balance in a movement scheme is to take care that effectively you are compared the same number of times to all other pairs, or to put it better: with the same weights. This ideal is often not achievable. But we will see that in many cases existing movements may be changed by simple means, such that the balance is improved considerably.

Some misconceptions about the balance

It is more important whom you are compared to than whom you play against

Not true. If you score a top you get a topscore. If a pair you are compared to scores a top you don't get a 0 right away, but your score on that board only decreases somewhat. Only if all pairs you are compared to do better than you, you get a 0.
In a contest where the number of pairs is much larger than the number of rounds the deciding factor is whom you play against. Only in contests where the number of rounds is larger than, or equal to, half the number of pairs, a direct encounter may be completely compensated by playing several times in the same compass direction.

When every pair of pairs is compared the same number of times the movement is well balanced

Suppose all pairs of pairs play the same number of times in equal as well as in opposite compass directions, but each pair encounters only half of the number of pairs. Then there is absolutely no balance. On the other side, if the players you don't meet, play the majority of boards in the same compass direction as you it may be possible that a reasonable balance is obtained. As we will see there is a way take such effects into account and make an effective assessment,

... to whom you are compared ...

This expression is often used loosely (as I did in the above), but the speaker sometimes is not aware that the effect of playing in equal compass directions is exactly as large (but in the other direction) as playing in opposite directions. We should of course take both into account. Furthermore the sum of the number of boards played in equal and opposite directions is not automatically the same for each pair of pairs. It makes a difference if they had a direct encounter.

The program balans

A program was developed for checking and improving the balance of movements in duplicate bridge. After a brief outline of the theory the implementation in program balans is described.
Finally we give some examples.

If you are not interested in technical details, skip to the Results page. Just keep in mind we use a quantity Qf with values between 0 and 100 as a measure of the balance where Qf = 100 is perfect.

Theory.

Call
P = number of pairs (even)
r = number of rounds

t = P/2 = number of times a board is played
b = number of boards per round
h = t-1 is half a top on a board
N = P * (P-1) /2 = number of pairs of pairs.
S = t * r * b * h = "amount of competition"

For simplicity we take b=1 (does not matter for the balance).
Calculate, for each of the N possible combinations of pairs, a score which is the sum of contributions of each board, as follows: The sum of these N scores is S

We further call SS = the sum of the squares of all these scores. It is this number we want to minimize in order to optimize the balance.

The method explained above is taken from http://chrisryall.net/memories/john.manning.htm
See also F.C. Schiereck, Groot Schemaboek N.B.B. Versie 2002

The Quality factor.

We compare SS with an estimated value for a perfect balance. The average score for a combination of pairs is S/N, and if each point had this value we would have
SS = N * (S/N)² or S2 / N.
Therefore we might define a Quality factor as
Qc = 100· S2

N·SS
For a perfect balance this number will be 100.

An modified version of the quality factor takes into account that the scores are integer values, so they cannot be exactly equal to S/N, since S/N is usually not an integer. We try to find a set of N integer numbers whose sum is S and and the sum of squares of which is minimal. This minimum of the sum of squares we use instead of S * S / N in the formula for Qc. We will use this modified version, called Qf, to be able to compare results with those of Schiereck. The difference between Qc and Qf is usually small.

Further remarks.

Another criterion for the balance, used by John Manning, is the spread of the scores around the average value, expressed as the standard deviation:
sd =

 

(SS −S2N)/ N
 
Differently from Qc in this case we have "the smaller the better". It is easy to show that the minimum of the sum of squares SS coincides with the minimum of the standard deviation and the maximum of Qc. The relation between Qc and sd is:
sd = S

N
·

 

(100 − Qc)/ Qc
 
One may represent the scores in a P by P matrix, the score matrix. It is a symmetrical matrix whose diagonal is undefined. Each element is the mutual score of two pairs.

Like Schiereck and Manning we use the word 'score' which may be confusing. A high 'score' for the combination pair1-pair2 means a good result for pair1 is bad for pair2 and vice versa. 'Correlation' might be a better word.

Why we use weights h and 1

PairContractRes.Score  MP  
NS EW NS EW
13 NT   C +600  3
2 - 1 -   -600 5
33 NT   C +600  3
4 - 3 -   -600 5
53 NT   C +600  3
6 - 5 -   -600 5
73 NT   +1 +630  8
8 - 7 -   -630 0
93 NT   C +600  3
10 - 9 -   -600 5
An example may make this clear. Suppose, on a certain board, the whole field bid and make 3NT, except at your table where opponents make an overtrick. Now you get a 0 instead of an average, a difference of half a top h. At another board the same opponents play in the same direction as you do, and again they are the only ones to make an overtrick. If all results on this board had been equal your MP score on this board would have been h, now it becomes h-1. The influence of the top obtained by opponents is now h times smaller.

See the scorecard shown beside. The average MP score is h=4. Since pair 7 scores a top all other pairs in the same compass directions score 1 MP less, and pairs in the opposite direction 1 MP more (except of course pair 8 who score 0).

In other words, for the influence on your result, playing against a pair once is equivalent to playing h times in the same direction as this pair.

Odd number of pairs.

The formalism described above may also be applied if the number of pairs is odd. Then we have t = (P-1)/2, but the rest is the same. In practice one usually uses a movement for an even number, 1 higher than the actual number, and considers one pair absent in this movement.

The vacancy quality.

The Quality factor does not say everything about the quality of a movement. Usually a multitude of solutions exist with the same Qf. Starting from version 7 of balans (2017) we go one step further and also systematically study the quality of a movement when one pair is absent. The average for all possible choices of the absent pair is found to be a good way to distinguish between movements with the same Qf. We discuss this topic on a separate page.

Higher moments.

Another way to distinguish between movements with the same Qf is to consider higher moments. By choosing the minimum of the 4th powers of all scores minus the average value:
S4 =

j 
( scorej − S/N )4

one may avoid extreme values in the score matrix that are compensated by a number of values close to the average
This does indeed further improve the balance in some cases, but, as a rule, is less effective than optimalisation of the vacancy quality.

Other models.

The present approach is based on the distribution of encounters and of playing in equal and opposite compass directions. There are other ways to investigate the quality of a movement. We mention a few, in order of complexity. These models have one thing in common. We look at the final scores, in MP or in percent, in a hypothetical contest, where the score on each board depends on a pre-determined way on the strength of the contestants. This kind of tests show that even a "perfect" movement with Qf = 100 is often not so perfect as one would wish.

Scoring across the field.

The balance when scoring across the field is discussed on a separate page.

Cross-IMPs scoring.

The ideas expounded here are also applicable to cross-IMPs scoring.
(Butler scoring we don't use of course for a serious contest)

Seeding.

A quite different technique to promote equal opportunity for all participants in a bridge event is seeding. In the improvement of the balance we try to achieve that each pairs effectively have the same number of encounters with all other pairs. By contrast, in seeding one tries to achieve that each pair encounters an equal amount of competition, but not necessarily from the same pairs.
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Implementation

The program "balans" seeks the optimal balance, based on a given movement by interchanging the directions NS and EW.
Early versions of "balans" used the following strategy.
First all tables are switched one by one. The switch that yields the largest improvement of the balance is kept. This process is repeated with this improved configuration, until no further improvement is found.
We then do the same thing with different starting positions found by letting a random number generator decide which tables are switched in the starting positions. Of course, at every improvement encountered the best movement up till now is kept.

Meanwhile Gerrit van der Velde has been so kind as to put his program at my disposal (see Historical development). The strategy followed here is borrowed from a method used in solid state physics to model processes such as crystallization. Here the tables are switched in a random order. Not only profitable switches are accepted, but every now and then also switches that decrease the balance. When and how many depends on a "temperature" parameter. By alternatingly decreasing and increasing the temperature you hope to escape from secondary minima and finally to reach the absolute minimum. Again the overall best movement is preserved.

It proved that, especially for large numbers of tables, this strategy is much more effective. Therefore an an adapted version of the main subroutine of this programma has been adopted in "balans" as of version 4. A number of large movements from the tables "Multisession movements" and "Scrambled Mitchells" have been improved as a result.

Version 7 of balans is a major revision, and is a joint venture of Ulrik Dickow and me. In this version the optimalisation of the vacancy quality is introduced, and at the same time the strategy described above was extended and fine-tuned by Ulrik, resulting in a much faster program.

Is there any guarantee the absolute optimum is found? No there is not. Of all possibilities to switch n out of N tables only a limited number is investigated. In practice, in most cases no improvement occurs after a few hundred iterations. But, certainly for large groups, there remains a chance of further improvements.

Command line.

The program is started in a console window by the command line:
   balans [options] <movement file>
Some options are (for a complete list see file README_en.html):
 -c   : check the movement, no optimalisation
 -s n : do optimalisation for n iterations
 -r n : keep row n fixed (= round n)
 -t n : keep column n fixed (= table n)
 -f n : keep the first n positions fixed
 -h   : show a short description of the options
The number of iterations is the number of trials for improvement. A few thousand is sufficient in most cases.

One often wants to keep the first round fixed. This may be done without loss of generality, unless the number of tables is larger than the number of board groups (more tables than rounds). In that case it is still possible without problem to fix as many tables in round one as there are board groups.
Option -f is specially for this case.

Application of keeping a column fixed: in a movement of 14 pairs, 6 rounds, one pair (14 or 13) always sits at table 7. For this pair it is convenient if also the direction NS or EW is always the same. Thus you may try if this is feasible without disturbing the optimal balance.

The normal output of the program contains also the quality factors for the case of an absent pair, for all choices of which pair is absent.

The lay-out of the movement file.

is as follows: Example (8 pairs, 6 rounds):
8 6 6 6 0
 1- 2 A  3- 4 B  5- 6 C  7- 8 D  0- 0 0  0- 0 0 
 3- 6 A  1- 7 B  0- 0 0  0- 0 0  5- 8 E  4- 2 F 
 7- 5 A  8- 6 B  1- 4 C  2- 3 D  0- 0 0  0- 0 0 
 4- 8 A  0- 0 0  0- 0 0  1- 6 D  2- 7 E  5- 3 F 
 0- 0 0  5- 2 B  3- 7 C  0- 0 0  6- 4 E  1- 8 F 
 0- 0 0  0- 0 0  8- 2 C  5- 4 D  1- 3 E  7- 6 F 
The pairs should be numbered starting with 1, the characters to designate board groups may be chosen arbitrarily. In the example above the number of tables is larger than half the number of pairs. Therefore not all tables are in use simultaneously. Tables not in use are indicated by zeroes.
For further possibilities click here

For an example of the output, click here.

The program in written in C, developed in the MinGW environment, and compiled with gcc.

Special thanks are due to Gerrit van der Velde for making his code available and to Joop van Wijk for critical comments and useful suggestions in all stages of this work.

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Results.

A few movements for 8, 10, 12, and 14 pairs, 6 rounds, popular in the Netherlands were compared. For those interested in the details see the Dutch version of this page.

TEAM movements

In a close collaboration, in which most of the work was done by Joop van Wijk, a set of movements was selected and improved, for contests of 4 to 16 pairs, and 6, 7 or 8 rounds. Maybe improvements are still possible here and there, but these are the optimal results of our search and calculations. We call this set the TEAM movements, and after an update in 2011 TeamPlus movements.

There are two types of movements, (Short) Howells and Scheveningen. In both cases the arrangement in the first round is the same, the so called universal starting position. Hence, the scores of the first round may be kept without change, whenever you want to switch to another movement after this round, e.g. due to absentees.

In the Howell movements pair 1 is always stationary, NS at table 1. You might want to use this for a person who has problems with walking or who needs special provisions. For the rest, players and boards move after every round. The number of tables is minimal. For 10 pairs you need only 5 tables, not 6 or 7.

In the Scheveningen movements all boards always remain on their original table, except if they have to be shared. In a movement of 7 rounds and 10 pairs therefore 7 tables are used. At appendix tables there always is a stationary pair.

We have paid attention to the requirement that a good balance should remain when one pair is absent.

In the following table Qf is the quality factor for an even number of pairs, Qf1 the quality factor if the highest pair number is absent.

T E A M   M O V E M E N T S

movement           Qf     Qf1    Howell  Sch
pairs  rounds

 6       6         83.33   80      y
 8       6         80      71.74   y      y See 8p6rGSBopt.asc for a better choice
10       6         83.76   75.79   y      y
12       6         79.75   71.81   y      y
14       6         70.41   70.36   -      y
16       6         76.92   71.92   -      y

 8       7        100     100      y      y
10       7         84.94   82.93   y      y
12       7         76.87   75.82   y      y
14       7         92      85.09   y      y
16       7         78.74   72.17   -      y
18       7         63.95   62.38   -      y

 8       8         87.50   86.36   y      -
10       8         89.51   82.50   y      y
12       8         85.38   78.33   y      y
14       8         82.46   78.15   y      y
16       8         94.64   88.64   y      y

Downloads

Via the download page you may download these movements in various formats.
File team_loopbrief.zip contains the movements in MS-Word format, and ready-made guide cards prepared by Joop.
Some movements not not listed above are included in teamplus_pakket.exe. i.e. movements for larger groups, for up to 24 pairs.
The file also contains a program to upload the movements into the scoring programs Bridge-It and NBB_Rekenprogramma. Uncheck "toevoegen aan NBB_R" if this is not wanted.

Acknowledgements.

These movements were collected and optimized by Joop van Wijk and me. It is not feasible to mention all original sources, such as Mr Howell and Mr Mitchell, Groot Schema Boek by F. Schiereck, Movements - a fair approach by Hallén, Hanner en Jannersten, the Internet ...
The constructive comments by Frans Lejeune and Dick Boogaers are appreciated. Bas Overwater put some programs at out disposal for processing movements and guide cards.

Multisession movements

The program balance is also very suitable to optimize a movement over several sessions. Preferably all pairs have had the same number of mutual encounters. To achieve this, one sometimes has to resort to sessions with different numbers of rounds. We also include a few where the the sessions are of equal length but the number of encounters varies.

A note of warning: only use this type of movement for events with a fixed group of participants. As a result of the optimization over multiple session the balance per session deteriorates, such that there is no point in using them for a variable number of participants.

pairs  rounds  sessions   rounds per session  encounters      Qf
     
  10     18        2            9               2           100     100      monster
  10     18        3            6               2           100     100      monster
  12     11        2         5 en 6             1           100     100      gedrocht
  12     11        3         3 en 4             1           100     100      gedrocht
  12     24        4            6             2  3          99.24   96.97   vals monster (nov. 2008)
  12     30        5            6             2  3          99.55   97.40   vals monster (nov. 2008)
  14     26        2           13               2           100     100      monster
  14     26        4         6 and 7            2            99.03  100      gedrocht     (nov. 2008)
  14     24        4            6             1  2          98.74   96.32   vals monster (nov. 2008)
  14     30        5            6             2  3          99.26   96.15   vals monster (nov. 2008)
  14     36        6            6             2  3          99.52   97.74   vals monster (nov. 2008)
  14     39        6         6 and 7            3            99.38  100      gedrocht
  16     15        2         7 and 8            1           100     100      gedrocht
  16     15        3            5               1           100     100      monster (Sep. 2007)
  16     30        5            6               2            99.12  100      monster (Sep. 2007)
  18     17        3         6, 6, 5            1            94.74  100      gedrocht
  20     19        3         6, 6, 7            1            96.10  100      gedrocht
  22     21        3            7               1            95.98  100      monster
  24     23        3         8, 8, 7            1            96.58  100      gedrocht

some of the above also appear in the section on "perfect movements" below.

The movements listed above are avialable on the downloadpage, click on "monster_pakket.exe". Again the uploading into Dutch scoringprograms may be turned off by unchecking "toevoegen aan NBB_R".

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Arrow Switching in Mitchell movements

In a Mitchell movement the field may be separated into NS and EW, each with their own winner. Besides the Standard Mitchell other types of movements with this property will be considered in this section. Such movements may be modified by "arrow switches" so as to obtain a balanced movement for a contest with one winner. We will limit ourselves to "complete" movements, where the number of rounds is equal to the number of tables, and all pairs play all boards.

Mitchell and variants

In the Standard Mitchell the movement from one round to the next is as follows:
- The NS pairs stay where they are
- The EW pairs go up one table
- The boards go down one table.
The going up and down is cyclic: up from the highest table means going to the lowest one and vice versa.

A simple variation on this theme is:
- The NS pairs go up one table
- The EW pairs go down one table
- The boards stay where they are.

Schemes like this work fine when the number of tables is odd. But for an even number of tables duplications occur after the first half the session. An effective solution to this problem is the Relay Mitchell. A disadvantage is that board sharing occurs in each round.

A better solution is the Double Weave Mitchell. The movement is more complicated but board sharing is avoided. However, this method only works when the number of tables is a multiple of 4, i.e. 4, 8, 12 ...

Another option for an even number of tables is the Skip Mitchell. Here the moving pairs do not meet all the stationary pairs. This inferior movement will not be considered further.

For a description of the above mentioned movements see "Movements - a fair approach" by Hallén, Hanner en Jannersten.

We will also look at some "Scheveningen" movements, taken from "Groot Schemaboek 2002" by F.C. Schiereck. Similar to the Mitchell, each NS pair meets only EW pairs as opponents. The boards are stationary, the NS pairs are labelled with odd numbers, and EW have even numbers. For an odd number of tables these movements are equivalent to the Standard Mitchell. For an even number board sharing is avoided in most cases as opposed to the Relay Mitchell.

Examples

Consider the following Mitchell for 14 pairs and 7 rounds.
 1- 8 A    2- 9 B    3-10 C    4-11 D    5-12 E    6-13 F    7-14 G   
 1-14 B    2- 8 C    3- 9 D    4-10 E    5-11 F    6-12 G    7-13 A   
 1-13 C    2-14 D    3- 8 E    4- 9 F    5-10 G    6-11 A    7-12 B   
 1-12 D    2-13 E    3-14 F    4- 8 G    5- 9 A    6-10 B    7-11 C   
 1-11 E    2-12 F    3-13 G    4-14 A    5- 8 B    6- 9 C    7-10 D   
 1-10 F    2-11 G    3-12 A    4-13 B    5-14 C    6- 8 D    7- 9 E   
 1- 9 G    2-10 A    3-11 B    4-12 C    5-13 D    6-14 E    7- 8 F   
A pleasure to the eye and also perfectly balanced, when you calculate a result for NS and EW separately. This is evident from the score matrix.
   \   7   7   7   7   7   7   0   0   0   0   0   0   0
   7   \   7   7   7   7   7   0   0   0   0   0   0   0
   7   7   \   7   7   7   7   0   0   0   0   0   0   0
   7   7   7   \   7   7   7   0   0   0   0   0   0   0
   7   7   7   7   \   7   7   0   0   0   0   0   0   0
   7   7   7   7   7   \   7   0   0   0   0   0   0   0
   7   7   7   7   7   7   \   0   0   0   0   0   0   0
   0   0   0   0   0   0   0   \   7   7   7   7   7   7
   0   0   0   0   0   0   0   7   \   7   7   7   7   7
   0   0   0   0   0   0   0   7   7   \   7   7   7   7
   0   0   0   0   0   0   0   7   7   7   \   7   7   7
   0   0   0   0   0   0   0   7   7   7   7   \   7   7
   0   0   0   0   0   0   0   7   7   7   7   7   \   7
   0   0   0   0   0   0   0   7   7   7   7   7   7   \

If you only want one winner this is not satisfactory. The problem may be solved by switching NS and EW at all tables during one round, a so-called arrow switch round. It does not matter which round is switched. For instance:
 1- 8 A    2- 9 B    3-10 C    4-11 D    5-12 E    6-13 F    7-14 G   
14- 1 B    8- 2 C    9- 3 D   10- 4 E   11- 5 F   12- 6 G   13- 7 A   
 1-13 C    2-14 D    3- 8 E    4- 9 F    5-10 G    6-11 A    7-12 B   
 1-12 D    2-13 E    3-14 F    4- 8 G    5- 9 A    6-10 B    7-11 C   
 1-11 E    2-12 F    3-13 G    4-14 A    5- 8 B    6- 9 C    7-10 D   
 1-10 F    2-11 G    3-12 A    4-13 B    5-14 C    6- 8 D    7- 9 E   
 1- 9 G    2-10 A    3-11 B    4-12 C    5-13 D    6-14 E    7- 8 F   
By this simple measure the quality factor Qf shoots up from 46.94 to no less than 92. The new score matrix becomes:
   \   3   3   3   3   3   3   4   4   4   4   4   4   0
   3   \   3   3   3   3   3   0   4   4   4   4   4   4
   3   3   \   3   3   3   3   4   0   4   4   4   4   4
   3   3   3   \   3   3   3   4   4   0   4   4   4   4
   3   3   3   3   \   3   3   4   4   4   0   4   4   4
   3   3   3   3   3   \   3   4   4   4   4   0   4   4
   3   3   3   3   3   3   \   4   4   4   4   4   0   4
   4   0   4   4   4   4   4   \   3   3   3   3   3   3
   4   4   0   4   4   4   4   3   \   3   3   3   3   3
   4   4   4   0   4   4   4   3   3   \   3   3   3   3
   4   4   4   4   0   4   4   3   3   3   \   3   3   3
   4   4   4   4   4   0   4   3   3   3   3   \   3   3
   4   4   4   4   4   4   0   3   3   3   3   3   \   3
   0   4   4   4   4   4   4   3   3   3   3   3   3   \

Do not make the mistake of switching two rounds. Then the balance collapses and Qf goes down to 49.64

How this case works out in imaginary contests with one, and also with two very strong pairs we will see further on.

For an even number of tables such a regular scheme is not possible and tricks are necessary to avoid that pairs meet the same opponents or the same boards twice. For 6 tables the Relay Mitchell looks as follows:

 1- 7 A    2- 8 B    3- 9 C    4-10 E    5-11 F    6-12 A 
 1-12 B    2- 7 C    3- 8 D    4- 9 F    5-10 A    6-11 B 
 1-11 C    2-12 D    3- 7 E    4- 8 A    5- 9 B    6-10 C 
 1-10 D    2-11 E    3-12 F    4- 7 B    5- 8 C    6- 9 D 
 1- 9 E    2-10 F    3-11 A    4-12 C    5- 7 D    6- 8 E 
 1- 8 F    2- 9 A    3-10 B    4-11 D    5-12 E    6- 7 F 
where table 1 and 6 share boards in all rounds. An alternative is the following movement where board sharing is limited to the last two rounds.
 1- 2 A    3- 4 B    5- 6 C    7- 8 D    9-10 E   11-12 F 
11- 6 A    7-12 B    1-10 C    9- 4 D    5- 8 E    3- 2 F 
 3- 8 A    9- 6 B    7- 4 C   11- 2 D    1-12 E    5-10 F 
 5- 4 A   11-10 B    3-12 C    1- 6 D    7- 2 E    9- 8 F 
 5-12 D    1- 8 B    9- 2 C    3-10 D   11- 4 E    7- 6 F 
 9-12 A    5- 2 B   11- 8 C    7-10 A    3- 6 E    1- 4 F 
This is the Scheveningen-12 on pag 5. of the Groot Schemaboek.
Again, these movements may be transformed into well-balanced movements with one winner, by arrow switching. By switching all tables except either table 1 or table 6 in any round of the Relay Mitchell, or one of the rounds 1, 2, 3, or 4 of the Scheveningen-12, Qf increases from 46.67 to 84. An arrow switch in one of the last two rounds is not as good but still yields Qf= 79.75. Also here switching two rounds is disastrous, Qf then becomes around 40.

We see that one single arrow switch round is optimal for complete Mitchells of 12 and 14 pairs. But this is not a general rule. For a Mitchell for 22 pairs and 11 rounds, for instance, one arrow switch (Qf=89.2) is still better than 2 (Qf=85.04), but further optimalisation leads to Qf=93.41.

In the following table we present Quality factors for Mitchells and Mitchell-like movements that are "scrambled" by arrow switches.

Quality factor for Mitchells and Scrambled Mitchells.

                                               revised April 2017
Nr of rounds Type  nr of arrow switch rounds                  best value
= tables             0       1       2            3

      5      M     46.00   67.65   31.08 (2 5)                 74.68 *)

      6      RM    46.67   79.75   42.86 (2 3)                 84.00 **)
      6      GSB   46.67   84.00   40.65 (2 6)                 84.00

      7      M     46.94   92.00   49.64 (2 7)                 92.00

      8      RM    47.32   91.38   60.92 (2 7)                 91.38
      8      DWM   47.32   94.64   57.61 (2 5)                 94.64 ***)
      8      GSB   47.32   94.64   55.21 (2 8)                 94.64

      9      M     47.53   93.90   70.00 (2 7)                 93.90

     10      RM    47.78   90.03   78.93 (2 10)                93.80
     10      GSB   47.78   91.88   77.12 (2 10)                93.80

     11      M     47.93   89.23   85.04 (2 8)                 93.41

     12      RM    48.11   85.55   89.58 (2 12) 67.08 (2 3 12) 94.00
     12      DWM   48.11   86.59   89.86 (2 4)  65.24 (2 4 8)  94.00
     12      GSB   48.11   86.59   89.86 (2 12) 65.24 (2 3 12) 94.00

     13      M     48.22   84.02   92.97 (2 8)  72.99 (2 7 8)  93.96

     14      RM    48.35   81.08   94.27 (2 5)  79.06 (2 5 6)  94.77
     14      GSB   48.35   81.70   94.44 (2 8)  77.36 (2 8 11) 94.77

     15      M     48.44   79.56   95.49 (2 9)  83.03 (2 8 9)  95.49

notes.

M =   Standard Mitchell
RM =  Relay Mitchell
DWM = Double Weave Mitchell
GSB = "Scheveningen" movement from "Groot Schemaboek 2002" by F.C. Schiereck
The numbers in between brackets denote the switch rounds used in the
calculation.

*)The optimum balance for the 5-table Mitchell is obtained by switching
one round, for 3 or 4 tables.
**)The optimum balance for the 6-table Mitchell is obtained by switching
one round, for all tables except either table 1 or 6 (the sharing tables).
***)The optimum Qf for the 8-table DWM is obtained by arrow switching any round,
but only rounds 1,4,5,8 give also optimum vacancy quality.

Discussion

Only when the number of rounds is 12 or higher it is worthwhile to switch two rounds.

After all, the fact that one arrow switch is usually necessary and sufficient turned out to be nothing new.

The example of the 7-table Mitchell we started this paragraph with is borrowed from John Manning, who remarks about the result: "The standard deviation works out at 1.05 and cannot be further reduced by switching more or fewer boards".

Already in 1979 John Manning paid attention to the problem of the optimum number of arrow switches and indicated "A rough and ready rule is to switch about one eighth of the boards in a Mitchelltype movement."

John Probst investigated this problem mathematically and generally and comes to the same conclusion:

We must arrow switch slightly more than 1/8 of the rounds for fairness. Anything else is WRONG!!!

see:
J.Manning: The Mathematics of Duplicate Bridge Tournaments
(Bulletin of Institute of Mathematics and its Applications Vol. 15, No. 8/9, August/September 1979, pp201 - 206)
J.Probst: http://www.blakjak.org/why_1in8.htm
See also: http://www.blakjak.org/lws_men1.htm
Ross Moore: "Too many arrow-switches spoil the balance" (1992) comes to the same conclusion.

Arrow Switching at the NBB

The fact that one arrow switch is better than two was apparently not known to the editors of the multiplex guide cards 1993 supplied by the NBB. In the 14-pair, 7 rounds movement two rounds are arrow switched. As we know this has Qf=49.6. Switching back round 2 or 3 makes this 92. Also the movements supplied with the NBB scoring program leave lots to be desired. The Scrambled Mitchell 12 and Scrambled Mitchell 14 have Qf's of 36.8 en 49.6, respectively. The first one is even worse than a non-switched Mitchell.

Arrow Switching in "Movements a fair approach"

In the well-known book by Hallén, Hanner and Jannersten "Movements - a fair approach" a good balance is defined as one where "any two pairs are compared on half the number of boards" (page 90). What is meant is that in the ideal case, any combination of two pairs should play an equal number of times in the same direction and in opposite directions. I think this is a fallacy that leads to way too many arrow switches. The authors ignore the fact that a pair does not meet all other pairs. As outlined above the weight of a direct encounter is h times as high as playing once in the same direction (h=half a top) *). Consequently, for a good balance, pairs that meet should play a lot less in the same direction than pairs that don't.
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The "one strong pair" model

Example: 12 pair Mitchell and Scrambled Mitchell

To illustrate the importance of balance we consider an extreme example. In a session of 12 pairs, 6 rounds, all pairs at all tables always play 50-50. Except pair 1 who always get a full top on every board. At the end of the evening they score a well-deserved 100%. Now you might hope that the remaining 500% will be equally distributed over the other 11 pairs, who therefore would get 45.45%. This might be the case if we could find a movement with a perfect balance.

But if the following Mitchell movement was used:

 1- 7 A    2- 8 B    3- 9 C    4-10 D    5-11 E    6-12 F   
 1-11 C    2- 7 F    3-10 E    4-12 B    5- 8 D    6- 9 A   
 1- 9 D    2-12 C    3- 8 A    4- 7 E    5-10 F    6-11 B   
 1-12 E    2-10 A    3-11 F    4- 8 C    5- 9 B    6- 7 D   
 1- 8 F    2- 9 E    3- 7 B    4-11 A    5-12 A    6-10 C   
 1-10 B    2-11 D    3-12 D    4- 9 F    5- 7 C    6- 8 E   
the result looks as follows:
 pair  1 100.00
 pair  2  40.00
 pair  3  40.00
 pair  4  40.00
 pair  5  40.00
 pair  6  40.00
 pair  7  50.00
 pair  8  50.00
 pair  9  50.00
 pair 10  50.00
 pair 11  50.00
 pair 12  50.00
As we see the other pairs score either 40 or 50%. The pairs that were beaten by pair 1 still get 50% while those who did not even meet this pair get only 40%.

We now improve the movement as follows. In round 3 we do an arrow switch except at table 1.

 1- 7 A    2- 8 B    3- 9 C    4-10 D    5-11 E    6-12 F   
 1-11 C    2- 7 F    3-10 E    4-12 B    5- 8 D    6- 9 A   
 1- 9 D   12- 2 C    8- 3 A    7- 4 E   10- 5 F   11- 6 B   
 1-12 E    2-10 A    3-11 F    4- 8 C    5- 9 B    6- 7 D   
 1- 8 F    2- 9 E    3- 7 B    4-11 A    5-12 A    6-10 C   
 1-10 B    2-11 D    3-12 D    4- 9 F    5- 7 C    6- 8 E   
Then the result becomes:
 pair  1 100.00
 pair  2  43.33
 pair  3  43.33
 pair  4  43.33
 pair  5  43.33
 pair  6  43.33
 pair  7  46.67 
 pair  8  46.67
 pair  9  50.00
 pair 10  46.67
 pair 11  46.67
 pair 12  46.67
Pair 1 plays exactly the same boards against the same opponents as in the original movement. But now the results are much closer together. The deviations from the ideal value 45.45 are roughly twice as small. These smaller fluctuations are the result of the improved balance. The original movement has quality factor Qf=46.67, standard deviation sd=2.99, the second one Qf=84, sd=1.29.

We next show the disastrous effect of too much switching:

movement with 2 arrow switches:

 1- 7 A    2- 8 B    3- 9 C    4-10 D    5-11 E    6-12 F   
 1-11 C    2- 7 F    3-10 E    4-12 B    5- 8 D    6- 9 A   
 9- 1 D   12- 2 C    8- 3 A    7- 4 E   10- 5 F   11- 6 B   
12- 1 E   10- 2 A   11- 3 F    8- 4 C    9- 5 B    7- 6 D   
 1- 8 F    2- 9 E    3- 7 B    4-11 A    5-12 A    6-10 C   
 1-10 B    2-11 D    3-12 D    4- 9 F    5- 7 C    6- 8 E
Again we assume pair 1 always scores a top and on all other tables an equal average result is obtained. With this movement we get the result:
  
 pair  1 100.00
 pair  2  53.33
 pair  3  53.33
 pair  4  46.67
 pair  5  53.33
 pair  6  46.67
 pair  7  50.00   
 pair  8  36.67
 pair  9  43.33
 pair 10  36.67
 pair 11  36.67
 pair 12  43.33
We can not eliminate all accidental factors in duplicate bridge. When your opponents are the only ones to bid and make a cold slam there is nothing you could have done about it, yet you get a 0. But accidental factors as a result of a poor balance may be reduced considerably by using suitable movement schemes.

Tips:

One more Example, 14 pairs.

The same story for the Mitchell for 14 pairs and 7 rounds we considered above. We give the scores for a session where pair 1 always gets the top, and other pairs always average, right away in tabular form:
                      
arrow switches   none      1         2         3                
Qf              46.94     92        49.64     32.71

pair 1         100.00    100.00    100.00    100.00
pair 2          41.67     46.43     46.43     46.43  
pair 3          41.67     46.43     51.19     51.19   
pair 4          41.67     46.43     51.19     55.95  
pair 5          41.67     46.43     51.19     55.95
pair 6          41.67     46.43     51.19     51.19  
pair 7          41.67     46.43     46.43     46.43
pair 8          50.00     45.24     45.24     40.48
pair 9          50.00     45.24     40.48     40.48
pair 10         50.00     45.24     40.48     40.48  
pair 11         50.00     45.24     40.48     40.48  
pair 12         50.00     45.24     45.24     45.24  
pair 13         50.00     45.24     45.24     40.48  
pair 14         50.00     50.00     45.24     45.24
Again, we see in column 1 the perfect balance, as long as we work out separate scores for the two groups NS and EW. But for a joint result this is of course no good. In column 2, one arrow switch, the results of pairs who played against pair 1, and those that played in the same direction, are nicely and closely together with one single exception. In columns 2 and especially 3 we see again the effect of overcompensation. For completeness' sake: the arrow switches in the above are in rounds 2, 2+3, 2+3+4, respectively.

The "2 strong pair" model

For an example of the result of arrow switches, in a contest of 2 strong pairs in a field of average pairs click here

Perfect movements

Do movements with perfect balance exist? Yes, the Howells with an even number of tables may reach this.
Thus a Howell-4 with 4 pairs and 3 rounds, Howell-8 with 8 pairs, 7 rounds, a Howell-12, with 12 pairs and 11 rounds, a Howell-16 with 16 pairs and 15 rounds et cetera.

In some cases it is even possible to split the movement into two or more sessions, each with their own board sets, and still maintain perfect balance.

For an odd number of tables a perfect balance may be sometimes be obtained when the number of board sets is doubled. An example is the Howell-6 with Qf=100 given in "Movements a Fair Approach". This movement has Qf1av= 66.67, Qf1max= 100. A considerable improvement is found by optimizing the "vacancy quality". With program "balans" one finds in no time a superperfect movement with Qf1av=100. However now one has an arrow switch in mid-round in a few cases. In this superperfect variant this occurs for everyone, in only one round.

A survey of some good-quality movements is given below.

Some movements with perfect balance

 4 pairs  3 rounds  1 session   Qf=100
 4 pairs  6 rounds  1 session   Qf=100   same pairs meet twice
 6 pairs  5 rounds  1 session   Qf=100   2 board groups per round
 8 pairs  7 rounds  1 session   Qf=100   or this alternative
10 pairs  9 rounds  1 session   Qf=100   2 board groups per round
10 pairs 18 rounds  2 sessions  Qf=100
10 pairs 18 rounds  3 sessions  Qf=100
12 pairs 11 rounds  1 session   Qf=100
12 pairs 11 rounds  2 sessions  Qf=100
12 pairs 11 rounds  3 sessions  Qf=100
14 pairs 13 rounds  1 session   Qf=100   2 board groups per round
14 pairs 26 rounds  2 sessions  Qf=100
16 pairs 15 rounds  1 session   Qf=100
16 pairs 15 rounds  2 sessions  Qf=100
Some of these movements, in particular the Howell-10's are due to Joop van Wijk.

Perfect and Superperfect Movements

The behaviour of movements with a perfect balance in a contest with 2 strong pairs is interesting enough to devote a special page to. We find that there are still gradations in perfection. See Perfect and Superperfect Movements.

Case Histories

Individual Movements

In the selection of movements for individual contests quite different considerations apply, apart from the balance. See the individuals page.

Historical

This paragraph is probably most incomplete. All information on the historical aspects of the balance in pairs movements is welcome at my e-mail address.

The concept of "balance" probably is almost as old as competitive bridge. De mathematical foundation of the work on balance and arrow switches presented here was laid by John Manning (1979) in an article with the title The Mathematics of Duplicate Bridge Tournaments.

In the book : "Movements - a fair approach" (1994) one finds discussions that might have lead to same the model outlined above, but as noticed, in a practical application the authors miss the right track. Apparently the work of Manning was unknown to the authors.

Paul Vermaseren developed, in the 80's, an effective computer programme for the optimalisation of Howell schemes. The results did not get the attention they deserved. A few perfectly balanced "Howells in several sessions" appeared in Wekowijzer. The Howell for 16 pairs in 3 sessions given above is one of those results.

The "least squares" approach used here I encountered in the Groot Schema Boek by Schiereck (2002) and on John Manning's web page. Both of these developments apparently took place independently. The person chiefly responsible for the results in the Groot Schema Boek is Gerrit van der Velde, who already in the 90's developed a computer programme to optimize the balance. Parts of this programme are used in "balans".

The "1 in 8" rule for the optimal number of arrow switches in Mitchell Movements is found already in the article of Manning (1979) cited above. It was further propagated by the work of John Probst on this subject. The least squares approach to characterize the balance, and recommendations about the optimal number of arrow switches were also given by Ross Moore (1992).

However there seem to be many places on earth where these results are still largely unknown.

The idea of "vacancy quality", and its use for a further improvement, rose in discussions with Ulrik Dickow and is introduced for the first time in the current work.

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